Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Inertia groups generate Galois Group

While reading a paper about the Kronecker-Weber Theorem, I noticed a theorem saying that for a Galois extension $K/\mathbb{Q}$, its Galois group is generated by $I_p$s, being the inertia groups of ...
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$\mathbb{Q}(\sqrt{1-\sqrt{2}})$ is Galois over $\mathbb{Q}$

I am trying to show that $E = \mathbb{Q}\left(\sqrt{1-\sqrt{2}}\right)$ is galois over $\mathbb{Q}$. The extension has the minimal polynomial ...
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1answer
35 views

Inclusion among subextensions of a cyclic field extension

I have some troubles with this problem: Let $E/F$ be a Galois extension with Galois group cyclic. Prove that two intermediate $B_1$ and $B_2$ satisfy $B_1\subseteq B_2$ if $[E:B_2]$ divides ...
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Question about field norm from Dummit and Foote 14.2.17(d)

I'm trying to solve the last part of an exercise in Dummit and Foote. Let $K/F$ be any finite separable extension, and let $\alpha\in K$. Let $L$ be a Galois extension of $F$ containing $K$ and ...
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Normal extension of any degree

Could any one tell me how to show that for any positive integer n, there exists a normal extension of rational number field of degree n?
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What is the Galois group of the splitting field of $X^8-3$ over $\mathbb{Q}$?

I've computed the splitting field of $x^8-3$ over $\mathbb{Q}$ to be $\mathbb{Q}(\sqrt[8]{3},\zeta_8)=\mathbb{Q}(\sqrt[8]{3},\sqrt{2},i)$, which is of degree 32 over $\mathbb{Q}$. The possible ...
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Chain of fields that are Galois over all subfields

Is there an example of fields $F_1$, $F_2$, and $F_3$ such that $\mathbb{Q}\subset F_1\subset F_2\subset F_3$ such that $[F_3:\mathbb{Q}]=8$ and each field is Galois over all its subfields but $F_2$ ...
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Galois extensions of degrees $p$ and $p^{n-1}$ given a Galois extension of $p^n$

Suppose $K$ is a Galois extension of a field $F$ of degree $p^n$ for a $p$ a prime. I want to see if there are Galois extensions of degrees $p$ and $p^{n-1}$ over $F$. If $G=\text{Gal}(K/F)$, then ...
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726 views

Fastest way to compute subfields of $\mathbb{Q}(\sqrt[8]{2},i)$ which are Galois over $\mathbb{Q}$?

I have the lattice of subfields of the splitting field $\mathbb{Q}(\sqrt[8]{2},i)$ over $x^8-2$, and the corresponding lattice of subgroups of the Galois group $G$ of the splitting field. I'm now ...
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201 views

Fixed field of Galois group of an infinite field $E$ is equal to $E$

I am trying to prove that if $E$ is an infinite field, then the fixed field of $Gal(E(x)/E)$ is $E$. The first part of the question was to find all automorphisms $$x\longmapsto ...
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Showing $E^H = K(\operatorname{Tr}_{E/{E^H}}(a))$

Suppose $E = K(a)$ and $E/K$ is a Galois extension. Show that if $H \leq \operatorname{Gal}(E/K)$ then the fixed field $E^H = K(\operatorname{Tr}_{E/{E^H}}(a))$. $K(\operatorname{Tr}_{E/{E^H}}(a)) ...
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725 views

a visual route to learning Galois theory

I really like the ideas of Galois theory: that you can think about all the algebraic numbers you can make starting with some set of them that there is some structure to this set of "algebraically ...
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Reading the mind of Prof. John Coates (motive behind his statement)

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as ...
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Norms for quadratic extensions

Let $K_2/K_1$ and $K_3/K_2$ be quadratic extensions of number fields. There is a homomorphism from $K_3^\star$ to $K_2^\star$ which sends an element $x$ to $N_{K_3/K_2}(x)^2/N_{K_3/K_1}(x)$. Any ...
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124 views

On Intermediate Fields of $\mathbb{C}(x_1,\dots,x_n)$

I am recently reading some Galois Theory, and a question occurred to me: What are the intermediate fields of $K$ of $\mathbb C(x_1,\dots,x_n)$, where $n$ is an arbitrary integer? I am aware of a ...
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267 views

Elements of $K$ which are separable over $F$ form a subfield of $K$?

I'm trying to prove the following statement: If $K$ is an extension of $F$ prove that the set of elements in $K$ which are separable over $F$ forms a subfield of $K$. I have a proof for the set ...
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1answer
616 views

Order of Galois group divides the degree of the extension

I keep seeing this theorem used in many textbooks but none of them provide proof (or there is no text layer so I can't find it!). Here is the statement in Algebra (Artin, pg. 540): (1.6) Theorem. ...
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When does the fraction field of a ring have a non-trivial Galois extension

I have read this previous question on existence of a non-trivial Galois extension. I was wondering about the following situation. Suppose, $R$ is a domain that is not a field. When does the fraction ...
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How do I find a splitting field $x^8-3$ over $\mathbb{Q}$?

Here's the situation. I am in this algebra class, and so far we have defined splitting fields and proved their existence and uniqueness. We have not yet decided on any rigorous definition of complex ...
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Splitting of primes in Galois extensions

I have the following problem: suppose $F/K$ is an abelian (Galois) extension of number fields, Galois group G, and $\mathfrak{p}$ is a prime of K, $\mathfrak{P}$ a prime of F dividing $\mathfrak{p}$; ...
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Uniqueness of splitting fields

I am reading a set of Galois theory notes, and I don't understand the proof of the following theorem: Let $f(X)$ be in $K[X]$, let $L/K$ be a splitting field for $f(X)$. Let $s\colon K\to M$ be ...
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Show If $\operatorname{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$ then $\mathbb{i} = \sqrt{-1} \notin K$

I'm troubled by solving a homework problem: If $\operatorname{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$ then $\mathbb{i} = \sqrt{-1} \notin K$ Any hints?
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Various question about Galois groups

I am presently learning the Galois theory, and is often confused when putting what I have learned to applications to polynomials. The following are some examples. (a) Every polynomial has a Galois ...
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reducible polynomial modulo every prime

how to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$. For example i know that $\mod 2 $, $x^4+1=(x+1)^4$ . Also $\mod 3$,we have that $0,1,2$ are not ...
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Example of matrix $M\in GL_3(\mathbb{Z}/7\mathbb{Z})$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is 3

I would want to make an example of a matrix $M \in GL_3(\mathbb{Z}_7)$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is $3$. Any hints how to do that ...
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1answer
127 views

For which $k$ do the $k$th powers of the roots of a polynomial give a basis for a number field?

Let $f \in \mathbb{Q}[x]$ of degreee $d$ be irreducible, with roots $\alpha_1,\ldots, \alpha_d$. One particular basis for the field extension of $\mathbb{Q}$ obtained by adjoining the roots of $f$ is ...
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144 views

$H^1$ with values in $\mathbb Q/\mathbb Z$

Given a field $k$, how do I have to think about the Galois cohomology group $H^1(k,\mathbb{Q}/\mathbb{Z})$? I know this is the group of continuous homomorphisms from $\Gamma_k$ to ...
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Finding Polynomials of Intermediate Galois extensions

Let $G$ be the Galois group of an irreducible polynomials $f(x)$ in $\mathbb{Q}[x]$. Let $K$ be the splitting field of $f(x)$. From the fundamental theorem of Galois theory we have that the ...
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Quartic extensions

Let $k$ be a perfect field and let $K/k$ be a quartic extension. Let $L/k$ be the normal (= Galois) closure of $K/k$. What structure can the Galois group $\text{Gal}(L/k)$ have? Of course it can be ...
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Galois Group over Ring of Integers

Suppose we have a quadratic (Galois) extension of $\mathbb{Q}$, call it $k$ with Galois group $G$. If we look at the ring of integers inside of $k$, call it $\mathcal{O}_k$, is it true that ...
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238 views

Local Langlands correspondence: Weil-Deligne group

While reading the book 'Langlands correspondence for loop groups', I came across the definition of the Weil group $W_F$ and the Weil-Deligne group $W'_F = W_F \ltimes \mathbb{C}$ with action ...
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Dimension of compositum of field extensions

Suppose we have fields $L$, $M$ and $N$ all infinite algebraic Galois extensions of a field $k$ such that $L \cap M$, $L \cap N$ and $N \cap M$ are finite dimensional extensions of $k$. Then is $L ...
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Showing that $[\mathbb C(z):\mathbb C(f_0)]=$topological degree of $f_0\in\mathbb C(z)$ as a function from the Riemann sphere to itself

McKean & Moll offer the following sketch of a proof. Let $P(x)=a_0(x)-f_0b_0(x)$ in $\mathbb C(f_0)[x]$ where $f_0(z)=\dfrac{a_0(z)}{b_0(z)}$ is in $\mathbb C(z)$. Then evidently, $\deg P=\#$ ...
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Does product of Galois groups equal to the Galois group of corresponding fields intersection?

Let $k$ be a field, $\bar{k}/k$ be a Galois extension with $G=Gal(\bar{k}/k)$ be an Abelian group(may be infinite). If $K,L$ are intermediate fields, denote $G_K=Gal(\bar{k}/K), G_L=Gal(\bar{k}/L)$. ...
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1answer
340 views

Does every field have a non-trivial Galois extension?

Clearly a field $F$ with no Galois extensions must have only non-separable elements in any extension (otherwise, take the minimal polynomial of some separable element over $F$ - its splitting field ...
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Subgroups of groups of order $2^{a-1}$

The context here is the following exercise Let $m=2^a$ with $a > 2$. Show that $\mathbb{Q}(\theta_m)$ contains exactly three quadratic subfields. By Galois theory, this reduces to the problem ...
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About cyclotomic extensions of $p$-adic fields

I've been working on the problem of finding the maximal abelian extension of $\mathbb{Q}_5$ that is killed by $5$. In other words, find the abelian extension of $\mathbb{Q}_5$ with Galois group ...
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Field $\mathbb{Q}(\sqrt{3}, \sqrt{-1})$

Am I right to say that the field $\mathbb{Q}(\sqrt{3}, \sqrt{-1})$ is an algebraic extension of $\mathbb{Q}$? Because $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt{3})( ...
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Are there (finite) field extensions which aren't Galois?

Let $K=F(\alpha)$. It strikes me that there are two possibilities: $\alpha \in F$, in which case $K\cong F$ $\alpha\not\in F$, in which case $K$ is the splitting field of $(x-\alpha)$ (and hence ...
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Degree of a field extension when compared to a Galois group

Consider $K=Q(\sqrt{2},\sqrt{3})$ - I think $K$ is Galois since it's the splitting field of $(x-\sqrt 2)(x+\sqrt 2)(x-\sqrt 3)(x+\sqrt 3)$. I feel like $G(K/F)$ is isomorphic to the Klein 4 group ...
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The radical solution of a solvable 17th degree equation

(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients: $$\begin{align*} x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 ...
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Galois group of $x^6+3$ over $\mathbb Q$

I'm having some difficulties finding the Galois group of the polynomial $g(x)=x^6+3$ over $\mathbb Q$. Here's what I did : I observed that the roots of the given polynomial are $\sqrt[6]3 ...
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A simple question about Galois groups

When talking about field extensions of degree two I understand the automorphisms in the Galois group intuitively as analogous to complex conjugation. I lose my understanding when going to field ...
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139 views

Galois Theory Problem

Let $L$ be a Galois extension of fields such that $Gal(L/K)=GL_{2}(\mathbb{F}_{p})$. Let $L_{1}$ and $L_{2}$ be subfields of $L$ containing $K$ corresponding to subgroups ...
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1answer
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Double root mod p implies transposition in galois group?

Let $K$ be the splitting field of some irreducible polynomial $f(x)$ in $\mathbb{Z}[x]$, and let $B$ be the integral closure of $\mathbb{Z}$ in $K$. Suppose that, for some prime $p$: $f(x) \equiv ...
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Intuition behind looking at permutations of the roots in Galois theory

What I find after reading books is that they explain only the conceptual definition and no one mentions the explanation behind it; I have been reading the Galois theory as many people told me to read, ...
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116 views

Classification of field extensions

Is it true that all extension of field $k$ are subfields of $\bar k$ (algebraic closure of $k$)? The same question for all algebraic extension. Thanks.
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Action of Galois group on set of generators

Let $k\subset K$ be a Galois extension, i.e. $$K=\langle{\xi}_{1}, {\xi}_{2},\ldots, {\xi}_{n}\rangle,$$ $$k\supset K=\{a_0+a_1{\xi}_{1}+\ldots+a_n{\xi}_{n}|a_0,\ldots,a_n\in k\}.$$ Is it true that ...
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1answer
108 views

Do we need to distinguish between positive and negative square roots?

I am seeking resolution of a disagreement over the nature of Galois theory. In the followup comments on a separate thread, I made the claim that the beauty of Galois theory is that it does not require ...
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Did Galois show $5^\sqrt{2}$ can't solve a high-order integer polynomial?

Suppose you have an unlimited quantity of the number one, and the operators plus, minus, multiply, divide, and power. Consider the (countable) set $S$ you generate by combining these: Using just one ...